Calculus [172 marks]

1. [Maximum mark: 7] 22N.1.AHL.TZ0.7

Consider the curve with equation (x 2
+ y )y
2 2
= 4x
2
where
x ≥ 0 and −2 < y < 2.

Show that the curve has no local maximum or local minimum points
for x > 0. [7]

2. [Maximum mark: 20] 22N.1.AHL.TZ0.10

The function f is defined by f (x) = cos
2
x − 3 sin
2
x, 0 ≤ x ≤ π.

(a) Find the roots of the equation f (x) = 0. [5]

(b.i) Find f ′(x). [2]

(b.ii) Hence find the coordinates of the points on the graph of

y = f (x) where f ′(x) = 0. [5]

(c) Sketch the graph of y = |f (x)|, clearly showing the

coordinates of any points where f ′(x) = 0 and any points
where the graph meets the coordinate axes. [4]

(d) Hence or otherwise, solve the inequality |f (x)| > 1. [4]

3. [Maximum mark: 6] 22N.2.SL.TZ0.3

The function f is defined as f (x) = ln(xe + 1) − x , for 0
x 4
≤ x ≤ 2

. The graph of f is shown in the following diagram.

 The graph of f has a local maximum at point A. The graph intersects the x-axis
at the origin and at point B.
(a) Find the coordinates of A. [2]

(b) Find the x-coordinate of B. [1]

(c) Find the total area enclosed by the graph of f , the x-axis and
the line x = 2. [3]

4. [Maximum mark: 6] 22N.2.AHL.TZ0.8

The following diagram shows liquid in a round-bottomed glass flask,
which is made of a sphere and a cylindrical neck.
 Initially, the flask is empty. Liquid is poured into the flask at a rate of
2 cm s
3 −1
. You may assume that the liquid does not reach the
cylindrical neck.

The volume V cm
3
and the height h cm of the liquid in the flask
[6]
satisfy the equation

V = 5πh
2
−
1

3
πh 3 .

Find the rate of change of the height of the liquid in the flask at the
instant when the volume of the liquid is 200 cm 3 .

5. [Maximum mark: 5] 22M.1.SL.TZ1.5

Consider the curve with equation y = (2x − 1)e
kx
, where x ∈ R

and k ∈ Q.

The tangent to the curve at the point where x = 1 is parallel to the

line y = 5e x.
k

Find the value of k. [5]

6. [Maximum mark: 14] 22M.1.SL.TZ1.7

A function, f , has its derivative given by f ′(x) = 3x 2 − 12x + p, where
p ∈ R. The following diagram shows part of the graph of f ′.

The graph of f ′ has an axis of symmetry x = q.

(a) Find the value of q. [2]

The vertex of the graph of f ′ lies on the x-axis.

(b.i) Write down the value of the discriminant of f ′. [1]

(b.ii) Hence or otherwise, find the value of p. [3]

(c) Find the value of the gradient of the graph of f ′ at x = 0. [3]

(d) Sketch the graph of f ′′, the second derivative of f . Indicate

clearly the x-intercept and the y-intercept. [2]

The graph of f has a point of inflexion at x = a.

(e.i) Write down the value of a. [1]

(e.ii) Find the values of x for which the graph of f is concave-down.

Justify your answer. [2]
7. [Maximum mark: 14] 21N.1.SL.TZ0.9
Consider a function f with domain a < x < b. The following diagram shows

the graph of f ′, the derivative of f .

The graph of f ′, the derivative of f , has x-intercepts at x = p, x = 0 and

x = t . There are local maximum points at x = q and x = t and a local

minimum point at x = r.

(a) Find all the values of x where the graph of f is increasing.

Justify your answer. [2]

(b) Find the value of x where the graph of f has a local maximum. [1]

(c.i) Find the value of x where the graph of f has a local minimum.
Justify your answer. [2]

(c.ii) Find the values of x where the graph of f has points of

inflexion. Justify your answer.
 [3]

(d) The total area of the region enclosed by the graph of f ′, the
derivative of f , and the x-axis is 20.

Given that f (p) + f (t) = 4, find the value of f (0). [6]

8. [Maximum mark: 7] 21N.1.SL.TZ0.5

The function f is defined for all x ∈ R. The line with equation y = 6x − 1 is

the tangent to the graph of f at x = 4.

(a) Write down the value of f ′(4). [1]

(b) Find f (4). [1]

The function g is defined for all x ∈ R where g(x) = x

2
− 3x and

h(x) = f (g(x)).

(c) Find h(4). [2]

(d) Hence find the equation of the tangent to the graph of h at

x = 4. [3]

9. [Maximum mark: 8] 21N.2.AHL.TZ0.8

Consider the curve C given by y = x − xy ln(xy) where x > 0, y > 0.

(a) dy dy
Show that dx
+ (x
dx
+ y)(1 + ln(xy)) = 1.
[3]

(b) Hence find the equation of the tangent to C at the point where
x = 1. [5]
10. [Maximum mark: 7] 21M.1.SL.TZ1.5
2
Consider the functions f (x) = −(x − h) + 2k and g(x) = e
x−2
+ k

where h, k ∈ R.

(a) Find f ′(x). [1]

The graphs of f and g have a common tangent at x = 3.

(b) Show that h =

e+6
. [3]
2

(c) Hence, show that k = e +

e
2

. [3]
4

11. [Maximum mark: 16] 21M.1.SL.TZ1.8

Let y for x > 0.
ln x
= 4
x

(a) Show that

dy
=
1−4 ln x
5
. [3]
dx x

Consider the function defined by f (x) for x > 0 and its graph
ln x
4
x

y = f (x).

(b) The graph of f has a horizontal tangent at point P. Find the

coordinates of P. [5]

(c) Given that f ′′(x) =

20 ln x−9
6
, show that P is a local
x

maximum point. [3]

(d) Solve f (x) > 0 for x > 0. [2]

(e) Sketch the graph of f , showing clearly the value of the x-

intercept and the approximate position of point P. [3]
12. [Maximum mark: 9] 21M.1.SL.TZ2.5
Consider the function f defined by f (x) = ln(x
2
− 16) for x > 4.

The following diagram shows part of the graph of f which crosses the x-axis at
point A, with coordinates (a, 0). The line L is the tangent to the graph of f at
the point B.

(a) Find the exact value of a. [3]

(b) Given that the gradient of L is

1
, find the x-coordinate of B. [6]
3

13. [Maximum mark: 7] 21M.2.AHL.TZ1.9

Two boats A and B travel due north.

Initially, boat B is positioned 50 metres due east of boat A.

The distances travelled by boat A and boat B, after t seconds, are x metres and
y metres respectively. The angle θ is the radian measure of the bearing of boat B

from boat A. This information is shown on the following diagram.

 (a) Show that y = x + 50 cot θ . [1]

(b) At time T , the following conditions are true.

Boat B has travelled 10 metres further than boat A.

Boat B is travelling at double the speed of boat A.
The rate of change of the angle θ is −0. 1 radians per second.

Find the speed of boat A at time T . [6]

14. [Maximum mark: 20] 21M.2.AHL.TZ1.11

3x+2
The function f is defined by f (x) = 2
4x −1
, for x ∈ R, x ≠ p, x ≠ q.

(a) Find the value of p and the value of q. [2]

(b) Find an expression for f ′(x). [3]

The graph of y = f (x) has exactly one point of inflexion.

 (c) Find the x-coordinate of the point of inflexion. [2]

(d) Sketch the graph of y = f (x) for −3 ≤ x ≤ 3, showing

the values of any axes intercepts, the coordinates of any local
maxima and local minima, and giving the equations of any
asymptotes. [5]
2

The function g is defined by g(x) , for x .

4x −1 2
= ∈ R, x ≠ −
3x+2 3

(e) Find the equations of all the asymptotes on the graph of

y = g(x). [4]

(f ) By considering the graph of y = g(x) − f (x), or otherwise,

solve f (x) < g(x) for x ∈ R. [4]

15. [Maximum mark: 6] 20N.1.SL.TZ0.T_13

Consider the graph of the function f (x) .
2 k
= x −
x

(a) Write down f ′(x). [3]

The equation of the tangent to the graph of y = f (x) at x = −2 is

2y = 4 − 5x.

(b) Write down the gradient of this tangent. [1]

(c) Find the value of k. [2]

16. [Maximum mark: 5] 20N.1.AHL.TZ0.H_2

Find the equation of the tangent to the curve y = e
2x
–3x at the

point where x = 0. [5]

17. [Maximum mark: 15] 20N.1.AHL.TZ0.H_11
Consider the curve C defined by y 2 = sin (xy) , y ≠ 0.

(a) Show that

dy y cos (xy)
.
dx
=
2y−x cos (xy)
[5]

(b) Prove that, when

dy
= 0 , y = ±1. [5]
dx

(c) Hence find the coordinates of all points on C , for

dy
0 < x < 4π, where
dx
= 0. [5]

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